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2.7: Grand canonical ensemble and distribution
https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Essential_Graduate_Physics_-_Statistical_Mechanics_(Likharev)/02%3A_Principles_of_Physical_Statistics/2.07%3A_Grand_canonical_ensemble_and_distribution
\[\begin{align} &\ln W_{m, N} \propto \ln M=\ln g_{env}\left(E_{\Sigma}-E_{m, N}, N_{\Sigma}-N\right)+\ln \Delta E_{\Sigma}=S_{env}\left(E_{\Sigma}-E_{m, N}, N_{\Sigma}-N\right)+\text { const } \nonum... $\begin{align} &\ln W_{m, N} \propto \ln M=\ln g_{env}\left(E_{\Sigma}-E_{m, N}, N_{\Sigma}-N\right)+\ln \Delta E_{\Sigma}=S_{env}\left(E_{\Sigma}-E_{m, N}, N_{\Sigma}-N\right)+\text { const } \nonumber\\ &\approx S_{env}\left|E_{\Sigma}, N_{\Sigma}-\frac{\partial S_{env}}{\partial E_{env}} \right|_{E_{\Sigma}, N_{\Sigma}} E_{m, N}- \left. \frac{\partial S_{env}}{\partial N_{env}} \right|_{E_{\Sigma}, N_{\Sigma}} N +\text{ const. } \label{102}\end{align}$

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